# Full text of "Treatise On Analysis Vol-Ii"

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```18       XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

En has a denumerable basis (Um/J)m^0 of open sets. For each n ^ 1, let 93n be

n                                    oo

the set of elementary sets in E of the form f] U/c/)> y x f]   Efc, where/is

y=l               * = »+!

any mapping of {1, 2, ...,«} into N. Since N" is denumerable (1.9.3), the
Set 93n is at most denumerable (1.9.2). We have seen that the union 93 of the
sets Sn is a basis for the topology of E (12.5.3); since 33 is denumerable
(1.9.4), it follows that E is separable (3.9.4).

(12.5.9) The product of a denumerable family of compact metrizable spaces is
compact and metrizable.

Let (En) be a sequence of compact metric spaces. Since we know already

00

(12,5.8) that E = Yl En is metrizable, it is enough to show that every

71=1

sequence (xn) of points of E has a cluster value (3.16.1). We define by induc-
tion on m ^ 0 a family of sequences (x^\^ of points of E, as follows:
x^ = xn; for m ^ 1, the sequence (x^)n^ is a subsequence of the sequence
(jcJi7""1^! (in other words, there exists a strictly increasing mapping
(pm: N->N such that x<m) = x\$%\$) such that the sequence (pTmx^\^
converges in Em to a point am. This construction is possible because Em is
compact. Now consider the sequence (yn) in E, where yn = x^ (Cantor's
"diagonal trick"). If we put \]/n = (pn o cpn_1 o - • • 0 <pi9 we have yn = x^n(n)\
since ^(72) > ^M-i(« — 1) by definition, it follows that (yrt) is a subsequence of
(xn). Furthermore, for each m, the sequence (y^n^m is a subsequence of the
sequence (x(nm^n^i9 and therefore the sequence (prmjFn)n^w converges to am.
Hence so does the sequence (prm ;>,,)„ ^19 since it differs from the former by
only a finite number of terms. Hence the sequence (yn) converges to the
point a = (aj (12.5.6).

PROBLEMS

1.    Show that the product of two quasi-compact topological spaces (Section 12.3,
Problem 6) is quasi-compact.

2.    Show that a topological space E is Hausdorff if and only if the diagonal (1.4.2) of
E x E is closed in E x E.

3.    Let (Ea)x 6! be a family of arbitrary (not necessarily uniformizable) topological spaces,
and let E «'fl Ea. Show that the elementary sets, defined as in (12.5.3), form a basis

ael

of a topology on E. This topology is called the product of the topologies of the E«.
Generalize (12.5,2) to (12.5.7) to this situation. neigh-
```